Let us recall some definitions and notions about n-exponentially convex functions (see [8]).
Definition 1
A real-valued function \(h : I \rightarrow\mathbb{R}\) on an open interval \(I, I \subset\mathbb{R}\) is called n-exponentially convex in the Jensen sense if
$$\sum_{j,k =1}^{n} b_{j} b_{k} h \biggl(\frac{x_{j} + x_{k}}{2} \biggr) \geq 0 $$
for all \(b_{i} \in\mathbb{R}\) and all \(x_{i} \in I\), \(i=1,\ldots,n\).
A real-valued function \(h : I \rightarrow\mathbb{R}\) is n-exponentially convex on I if it is n-exponentially convex in the Jensen sense and continuous on I.
Remark 2.1
-
(i)
From the definition it is obvious that the set of all n-exponentially convex functions on I is a convex cone.
-
(ii)
It is less obvious that a product of any two n-exponentially convex functions on I is again of the same type (see [18]).
-
(iii)
n-exponentially convex functions are invariant under admissible shifts and translations of the argument, that is, if \(x\mapsto f(x)\) is n-exponentially convex, then \(x\mapsto f(x-c)\) and \(x\mapsto f(x/\lambda)\) are also n-exponentially convex functions.
Definition 2
A real-valued function \(h : I \rightarrow\mathbb{R}\) is exponentially convex in the Jensen sense if it is n-exponentially convex in the Jensen sense for all \(n\in\mathbb{N}\).
A real-valued function \(h : I \rightarrow\mathbb{R}\) is exponentially convex if it is exponentially convex in the Jensen sense and continuous.
Remark 2.2
Note that a positive real-valued function \(h : I \rightarrow\mathbb{R}\) is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense, that is,
$$b_{1}^{2} h(x)+ 2 b_{1}b_{2} h \biggl(\frac{x+y}{2} \biggr) + b_{2}^{2} h(y) \geq0 $$
for all \(b_{1}, b_{2} \in\mathbb{R}\) and \(x, y \in I\).
If h is 2-exponentially convex, then it is log-convex. The converse is true if h also is continuous. n-exponentially convex functions are not exponentially convex in general. For example, see [18].
We will use the following basic inequality of log-convex functions.
Lemma 2.3
If
\(\Phi: I \rightarrow\mathbb{R}\)
is log-convex, then for
\(r< s< t\) (\(r,s,t\in I\)),
$$ \bigl(\Phi(s)\bigr)^{t-r}\leq\bigl(\Phi(r) \bigr)^{t-s} \bigl(\Phi(t)\bigr)^{s-r}. $$
(3)
Proof
See [19], p.4. □
Let us give a few basic examples of exponentially convex functions; for details, see [18].
Example 2.4
-
(i)
\(f(x)=c\) is exponentially convex on \(\mathbb {R}\) for any \(c\geq0\).
-
(ii)
\(f(x)=e^{\alpha x}\) is exponentially convex on \(\mathbb {R}\) for any \(\alpha\in \mathbb {R}\).
-
(iii)
\(f(x)=x^{-\alpha}\) is exponentially convex on \((0,\infty)\) for any \(\alpha>0\).
Lemma 2.5
-
(i)
For
\(p>0\), let
\(\varphi_{p}:[0,\infty )\rightarrow \mathbb {R}\)
be defined by
$$ \varphi_{p}(x)=\frac{e^{px^{2}}}{p^{2}}. $$
Then
\(p\mapsto\varphi_{p}(x)\), \(p\mapsto\frac{d}{dx}\varphi_{p}(x)\), and
\(p\mapsto\frac{d^{2}}{dx^{2}}\varphi_{p}(x)\)
are exponentially convex on
\((0, \infty)\)
for each
\(x\in[0, \infty)\).
-
(ii)
For
\(p>1\), let
\(\phi_{p}:[0,\infty )\rightarrow \mathbb {R}\)
be defined by
$$ \phi_{p}(x)=\frac{x^{p}}{p(p-1)}. $$
Then
\(p\mapsto\phi_{p}(x)\), \(p\mapsto\frac{d}{dx}\phi_{p}(x)\), and
\(p\mapsto\frac{d^{2}}{dx^{2}}\phi_{p}(x)\)
are exponentially convex on
\((1, \infty)\)
for each
\(x\in[0, \infty)\).
Proof
(i) follows from parts (ii) and (iii) of Example 2.4 and Remark 2.1.
(ii) follows by similar arguments as in part (i) noting that \(x^{p}=e^{p\ln x}\). □
The next simple lemma will be useful in our applications.
Lemma 2.6
Let
\(f : [0, \infty) \rightarrow\mathbb{R}\)
be a convex function with
\(f'(0) = 0\). Then
f
is an increasing convex function.
Proof
If f is a convex function, then \(f'\) is nondecreasing. Since \(f'(0)=0\), we have \(f'(x)\geq0\), that is, f is an increasing convex function. □
We opt an elegant method (see [18]) of constructing n-exponentially convex functions and exponentially convex functions.
Consider the following functional acting on nondecreasing convex functions:
$$ f\mapsto\Omega(f) = \frac{1}{P_{n}}\sum ^{n}_{i=1}p_{i} f\bigl(\Vert x_{i}\Vert \bigr)- f \Biggl(\frac{1}{P_{n}}\Biggl\Vert \sum ^{n}_{i=1}p_{i}x_{i} \Biggr\Vert \Biggr). $$
(4)
From Theorem 1.1 it follows that \(\Omega(f) \geq0\).
Theorem 2.7
Let
\(f\mapsto\Omega(f)\)
be the linear functional defined by (4) and define
\(\Phi_{1}:(0,\infty)\rightarrow \mathbb {R}\)
and
\(\Phi_{2}:(1,\infty)\rightarrow \mathbb {R}\)
by
$$ \Phi_{1}(p)=\Omega(\varphi_{p}),\qquad \Phi_{2}(p)=\Omega(\phi_{p}), $$
where
\(\varphi_{p}\)
and
\(\phi_{p}\)
are defined in Lemma
2.5. Then:
-
(i)
The functions
\(\Phi_{1}\)
and
\(\Phi_{2}\)
are continuous on
\((0,\infty)\)
and
\((1,\infty)\), respectively.
-
(ii)
If
\(n\in \mathbb {N}\), \(p_{1},\ldots,p_{n}\in(0,\infty )\), and
\(q_{1},\ldots,q_{n}\in(1,\infty)\), then the matrices
$$\biggl[\Phi_{1} \biggl(\frac{p_{j}+p_{k}}{2} \biggr) \biggr] _{j,k=1}^{n}, \qquad \biggl[\Phi_{2} \biggl( \frac{q_{j}+q_{k}}{2} \biggr) \biggr]_{j,k=1}^{n} $$
are positive semidefinite.
-
(iii)
The functions
\(\Phi_{1}\)
and
\(\Phi_{2}\)
are exponentially convex on
\((0,\infty)\)
and
\((1,\infty)\), respectively.
-
(iv)
If
\(p,q,r\in(0,\infty)\)
are such that
\(p< q< r\), then
$$\begin{aligned}& \biggl( \frac{\sum_{i=1}^{n}p_{i}\exp(q\|x_{i}\| ^{2})-P_{n}\exp ( \frac{q\|\sum_{i=1}^{n}p_{i}x_{i}\|^{2}}{P_{n}^{2}} ) }{q^{2}P_{n}} \biggr)^{r-p} \\& \quad \leq \biggl( \frac{\sum_{i=1}^{n}p_{i}\exp(p\|x_{i}\| ^{2})-P_{n}\exp ( \frac{p\|\sum_{i=1}^{n}p_{i}x_{i}\|^{2}}{P_{n}^{2}} ) }{p^{2}P_{n}} \biggr)^{r-q} \\& \qquad {}\times \biggl( \frac{\sum_{i=1}^{n}p_{i}\exp(r\|x_{i}\| ^{2})-P_{n}\exp ( \frac{r\|\sum_{i=1}^{n}p_{i}x_{i}\|^{2}}{P_{n}^{2}} ) }{r^{2}P_{n}} \biggr)^{q-p}; \end{aligned}$$
if
\(u,v,w\in(1,\infty)\)
are such that
\(u< v< w\), then
$$\begin{aligned}& \biggl( \frac{\sum_{i=1}^{n}p_{i}\|x_{i}\| ^{v}}{v(v-1)P_{n}}-\frac{ (\Vert \sum^{n}_{i=1}p_{i}x_{i}\Vert )^{v}}{v(v-1)P_{n}^{v}} \biggr)^{w-u} \\& \quad \leq \biggl( \frac{\sum_{i=1}^{n}p_{i}\|x_{i}\| ^{u}}{u(u-1)P_{n}}-\frac{ (\Vert \sum^{n}_{i=1}p_{i}x_{i}\Vert )^{u}}{u(u-1)P_{n}^{u}} \biggr)^{w-v} \\& \qquad {}\times \biggl( \frac{\sum_{i=1}^{n}p_{i}\|x_{i}\|^{w}}{w(w-1)P_{n}}-\frac{ (\Vert \sum^{n}_{i=1}p_{i}x_{i}\Vert )^{w}}{w(w-1)P_{n}^{w}} \biggr)^{v-u}. \end{aligned}$$
Proof
(i) The continuity of the functions \(p\mapsto\Phi_{i}(p)\), \(i=1,2\), is obvious.
(ii) Let \(n\in \mathbb {N}\) and \(\xi_{j}, p_{j}\in \mathbb {R}\) (\(j = 1, \ldots, n\)). Define the auxiliary function \(\Psi_{1} : [0,\infty) \rightarrow \mathbb {R}\) by
$$ \Psi_{1}(x)=\sum_{j,k=1}^{n} \xi_{j}\xi_{k}\varphi_{\frac{p_{j}+p_{k}}{2}}(x). $$
Now \(\Psi_{1}'(0)=0\) since \(\frac{d}{dx}\varphi_{t}(0)=0\), and
$$\Psi_{1}''(x)=\sum _{j,k=1}^{n}\xi_{j}\xi_{k} \frac {d^{2}}{dx^{2}}\varphi_{\frac{p_{j}+p_{k}}{2}}(x) \geq0 $$
for \(x\geq0 \) by Lemma 2.5, which means, by Lemma 2.6, that \(\Psi_{1}\) is an increasing convex function. Now Theorem 1.1 implies that \(\Omega(\Psi_{1})\geq0\). This means that
$$\biggl[\Phi_{1} \biggl(\frac{p_{j}+p_{k}}{2} \biggr) \biggr]_{j,k=1}^{n} $$
is a positive semidefinite matrix.
Similarly, we can define an auxiliary function \(\Psi_{2}\) concluding that
$$\biggl[\Phi_{2} \biggl(\frac{q_{j}+q_{k}}{2} \biggr) \biggr]_{j,k=1}^{n} $$
is a positive semidefinite matrix.
(iii) and (iv) are simple consequences of (i), (ii), and Lemma 2.3. □
The following application in the probability is a consequence of the above theorem and gives an interesting connection between moments of discrete random variables.
Corollary 2.8
Let
\((V,\|\cdot\|)\)
be a normed space, and let
X
be a discrete random variable defined by
\(P(X=x_{i})=p_{i}\), \(x_{i}\in V\), \(p_{i}>0\), \(i=1,\ldots,n\), \(\sum_{i=1}^{n}p_{i}=1\). Then, for
\(1< j< k< m\),
$$\begin{aligned}& \bigl\{ \mathbb{E}\bigl[\Vert X\Vert ^{k}\bigr]-\bigl(\bigl\Vert \mathbb{E}[X]\bigr\Vert \bigr)^{k}\bigr\} ^{m-j} \\& \quad \leq C(j,k,m) \bigl\{ \mathbb{E}\bigl[\Vert X\Vert ^{j}\bigr]-\bigl(\bigl\Vert \mathbb{E}[X]\bigr\Vert \bigr)^{j}\bigr\} ^{m-k}\bigl\{ \mathbb {E}\bigl[\Vert X\Vert ^{m}\bigr]-\bigl(\bigl\Vert \mathbb{E}[X] \bigr\Vert \bigr)^{m}\bigr\} ^{k-j}, \end{aligned}$$
where
$$ C(j,k,m)=\frac{\binom{k}{2}^{m-j} }{\binom{j}{2}^{m-k}\binom{m}{2}^{k-j}}. $$
(5)
Theorem 2.7 also sets the following model.
Theorem 2.9
Let
\(I \subset\mathbb{R}\)
be an open interval, and
\(\Gamma= \{\eta_{t} | t \in I\}\)
be a family of continuous functions defined on
\(J\subseteq [0,\infty) \)
such that
\(\frac{d}{dx}\eta_{t}(0)=0\), \(t\in I\), and
\(t\mapsto\frac{d^{2}}{dx^{2}}\eta_{t}(x)\)
is
n-exponentially convex
I
for any
\(x\in J\). Consider the functional
\(f\mapsto\Omega(f)\)
given in (4). Then
\(t \mapsto \Omega(\eta_{t})\)
is an
n-exponentially convex function on
I.
Remark 2.10
Other features of Theorem 2.7 can be easily added in the previous theorem.